At this point there are some technical details in the paper which we are going to skip, but the main idea is to the fact that (or in this case even the weaker hypothesis center brunching would work) to prove that in our case is a density point iff is a “leaf density point” in both its center-stable and center-unstable leaves. Hence by accessibility from to , we can “push” the point along the us-path that joins to and induce that is a “leaf density point” in hence a density point in .

all points are density points of hence .

Note that here if we replace accessibility by essential accessibility, we still get .

Hence , either or

is essentially constant. is ergodic.

This establishes theorem B.

Let be the set of measure preserving diffeomorphisms on that are of class

Theorem A: Accessibility is open dense in the space of diffeomorphisms in with .

For any , let denote the set of points that’s accessible from

Let is open

Fact: with is and
where is accessible and and is integrable

Note that this actually requires some rather technical work which was done in the paper, here we skip the proof of this.

Let is open

It’s easy to see that is automatically open hence is compact.

Proposition: Let , the following are equivalent:

1) has non-empty interior

2) is open

3) has non-empty interior in

Proof: 1) 2) 3) 1)

Mainly by drawing pictures and standard topology.

Unweaving lemma: latex PH^r(M)$ with s.t. the distance between and is arbitrarily small, and is open.